Gas station without pumps

2020 March 28

Exponential and logistic growth

Filed under: Uncategorized — gasstationwithoutpumps @ 16:37
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I was just thinking that the current COVID-19 crisis provides a teachable moment for showing the advantage of semilogy plots (log scale on the y axis, linear on the x axis) for showing exponential data.  So I grabbed information from one of the many sources reporting the number of cases and number of deaths due to COVID-19 [] and plotted them in different ways.

Here is what the data looks like on a linear graph:

This shows the rapid growth of the cases, but is hard to project into the future. It is also hard to see whether the growth rate is changing or how the deaths are related to the cases.

My next step was to use a log scale and to fit exponential curves to the data (actually fitting the log of the model to the log of the data, to avoid being biased by just the most recent data). I fit the data from March 2 on, since there were small-number effects before then.

The doubling time is alarmingly short for both the number of cases and for the number of deaths. Of course, exponential growth models don’t work well when projected indefinitely far into the future—at some point the infection rate predicted by the model exceeds 100%, which makes no sense. A better model is a logistic model: A \frac{2^{(t-t_c)/\lambda}}{1+2^{(t-t_c)/\lambda}}, which has three parameters:
A, the eventual fraction of the population affected
t_c, the time at which half of A is affected
\lambda, the doubling time.

I did fits of the logistic-growth model with three different assumptions about the eventual fraction affected: everyone, 10%, and 1% of the population, then fit the other parameters.

Right now, we can’t distinguish between any of these models, but by April 15 we may be able to see which curve we are on. I’ve added data points for Santa Cruz County and for California, which are both about five days behind the national curve.

I made no attempt to make projections for the deaths. That number seems to be growing at a slower rate than the total number of cases, which probably indicates that recent testing has been uncovering a larger fraction of the actual cases than earlier testing, and that the growth rate for the actual number of cases is doubling about every 3.2 days, not every 2.4 days. If that is correct, we may see a slow-down of the number of cases that does not indicate saturation of the logistic model, but just testing catching up with the backlog of cases.


  1. Good post. I also liked these two videos from MinutePhysics ( and TomRocksMaths (

    Comment by Miguel F. Aznar — 2020 March 28 @ 19:47 | Reply

    • Both of those are good videos. The visualization in MinutePhysics is interesting, but it does require animation to be interpretable. The S-I-R model in the TomRocksMath video is a simple model that leads to logistic growth. The big questions are what fraction of the total population is susceptible and how fast does the number of infections grow. The TomRocksMath video emphasizes how important the contact ratio is—it is what determines how fast the disease spreads. He assumes that the entire population is susceptible, which may or may not be true—we probably won’t know that for another couple of months (when everyone who is susceptible gets sick). I don’t hold out much hope of the US getting the contact ratio down to the point where the growth rate is substantially slowed (unlike South Korea and China).

      Comment by gasstationwithoutpumps — 2020 March 28 @ 20:22 | Reply

  2. has some nice dynamic graphs and 3Blue1Brown (one of the very best YouTube channels for people interested in math) has a decent (actually not his best work, but still pretty good) set of visualizations ( of infection spread with a variety of mitigations. None of the models I’ve seen take into account the possibility that conferred immunity is short lasting. If that’s the case, then while flattening the curve may still be important for protecting hospitals, it might actually cause the pandemic to begin to repeat… Until there is a vaccine or cure.

    Comment by whatisron — 2020 March 29 @ 20:21 | Reply

    • The graphs are the ones in the MinutePhysics video that Miguel pointed to. It is easier to see them on the site, as you don’t have to pause and rewind the video to see the animation repeated.

      I’m not holding up much hope of either a vaccine or a cure at this point—the last attempt at a coronavirus vaccine (for SARS) was a bust, as it made the disease worse. And cures for viral illnesses have been very few.

      I think that there is still a reasonable chance that this will have as big an impact as the Spanish Flu of 1918.

      What I don’t understand is why the huge concern about ventilators, rather than just hospital beds. From what little I’ve seen (, once someone needs a ventilator, they only have about a 19% chance of surviving, if they get a ventilator.

      Comment by gasstationwithoutpumps — 2020 March 29 @ 20:58 | Reply

  3. […] is a followup on my post Exponential and logistic growth, which was intended mainly as as teaching opportunity for showing the value of log scales on […]

    Pingback by Best visualizations of the COVID-19 spread | Gas station without pumps — 2020 April 1 @ 10:03 | Reply

  4. One of my students just posted to a class Piazza group:

    Here’s two videos that also talk about the virus and the value of logscale by 3blue1brown

    “Exponential growth and epidemics”

    “Simulating an epidemic”

    Comment by gasstationwithoutpumps — 2020 April 2 @ 17:10 | Reply

  5. […] Exponential and logistic growth I showed how both exponential and logistic models fit the data available then (28 March), but we […]

    Pingback by Exponential and logistic models aren’t working | Gas station without pumps — 2020 April 24 @ 00:01 | Reply

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